Ivy manufactures its own food, so it's not a parasitic plant, but it can cause problems if allowed to grow up trees. Ivy clings by attaching little pads to an upright support. These pads can crack or split the bark of a tree. Part of the problem with ivy encircling a tree is not so much that it will girdle and strangle the tree, but that the evergreen foliage traps and holds moisture, keeping the bark of the tree damp, which can soften it, and lead to disease. Ivy can also become heavy, especially when wet, putting additional strain on the tree. With all of these factors, you might want to reconsider allowing the ivy to grow up the tree, or at least keep it trimmed back and under control.